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Research Papers

In Situ Deformation of Silicon Cantilever Under Constant Stress as a Function of TemperatureOPEN ACCESS

[+] Author and Article Information
Ming Gan

School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907
e-mail: ganm@purdue.edu

Yang Zhang

School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907
e-mail: zhan1076@purdue.edu

Vikas Tomar

Associate Professor
School of Aeronautics and Astronautics,
Purdue University,
West Lafayette, IN 47907
e-mail: tomar@purdue.edu

1Corresponding author.

Manuscript received April 13, 2014; final manuscript received June 15, 2014; published online July 8, 2014. Assoc. Editor: Hsiao-Ying Shadow Huang.

J. Nanotechnol. Eng. Med 5(2), 021004 (Aug 19, 2014) (9 pages) Paper No: NANO-14-1035; doi: 10.1115/1.4027877 History: Received April 13, 2014; Revised June 15, 2014

Abstract

This research reports in situ creep properties of silicon microcantilevers at temperatures ranging from 25 °C to 100 °C under uniaxial compressive stress. Results reveal that in the stress range of 50–150 MPa, the strain rate of the silicon cantilever increases linearly as a function of applied stress. The strain rate (0.2–2.5 $×10-6s-1$) was comparable to literature values for bulk silicon reported in the temperature range of 1100–1300 °C at one tenth of the reported stress level. The experiments quantify the extent of the effect of surface stress on uniaxial creep strain rate by measuring surface stress values during uniaxial temperature dependent creep.

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Introduction

Microscale silicon structures have been an essential part of MEMS devices due to their excellent electrical and mechanical properties, as well as low manufacturing cost. Nanoscale silicon structures, e.g., silicon nanowires, have been used in devices such as field effect transistors, p-n diodes, and inverters [1]. The micro/nanoscale silicon structures are commonly subjected to mechanical stress when functioning, which may lead to creep deformation of the structures. When the working temperature increases, the creep deformation could cause permanent failure. Creep is a nonrecoverable plastic deformation occurring at low load regimes, constant stress, and small strain rates [2,3]. Creep occurs in the form of transient deformation of a material leading to permanent plastic strain at applied stress values lower than the material yield stress [2-4]. Creep of bulk materials starts with a decreasing strain rate in the primary (initial) stage followed by a steady state (the secondary stage) creep. After that, in the tertiary stage, the strain rate increases again until the material fails. Besides occurring at high temperature in bulk materials, creep deformation can also occur at nanoscale contacts at moderate temperature. Characterization of such nanoscale creep deformation is important for applications related to operation of miniature devices, thermal stability of interfaces etc.

The creep behavior of a material is expressed by a power law equation with stress exponent n usually used as an indicator of the creep mechanism [5]. It is believed that when the value of $n$ is 1, creep is controlled by vacancy diffusion [2,3]; when the $n$ value is 2, the creep mechanism is controlled by grain boundary sliding [6]; when $n$ is 3, diffusion-controlled dislocation motion dominates as deformation mechanism [7,8]; and when $n$ is 5, it is dislocation climb-controlled creep mechanism [9]. During microscale indentation creep tests on certain metals, alloys, and ceramics at room temperature, high stress exponent values up to hundreds have been observed [10-16]. The mechanism behind such high stress exponent values has been attributed to volumetric densification and dislocation pile up.

The creep properties of silicon have attracted research interest from the 1960s. Alexander and Haasen studied the creep of bulk silicon in 1968 [17]. Since then, research has been performed on silicon bulk samples [18-20], microscale structures [21,22], and nanowires [1]. For silicon, the brittle-to-ductile transition occurs at temperature between 520 °C and 600 °C [23]. Creep deformation of silicon is believed to be significant above this temperature range. Therefore, previous research mainly focused on creep deformation of silicon at the temperature above 600 °C. For example, Huff et al. studied the plastic deformation of a silicon capping layer in the temperature range of 900 °C–1200 °C [22]. Walters and Spearing characterized the creep properties of silicon in the temperature range of 600 °C–750 °C [20]. For most MEMS devices, the working temperature may not exceed 200 °C. Creep of silicon structures could still happen at low temperatures, especially at microscale or nanoscale. In this research, microscale creep behavior of silicon cantilevers is investigated from room temperature to 100 °C. The creep mechanism and the correlation between surface stress and creep strain rate is discussed.

Methods

In this work, the deformation of microscale silicon cantilever samples under constant load was investigated at 25 °C, 50 °C, and 100 °C, respectively. The load was applied by a modified mechanical loading platform, Fig. 1. The temperature was controlled by electronic heaters and monitored by resistance temperature detectors (RTD). An open-path Raman spectroscopy setup was integrated to the mechanical loading system to measure the near-surface stress of the sample during in situ deformation.

Experimental Setup.

The schematic diagram of the setup is shown in Fig. 1. As mechanical load was applied to the silicon cantilever in the uniaxial direction, the Raman laser was focused onto the side surface of the sample using a $40×$ objective. Back-scattered Raman signal was collected by the same objective and sent to a spectrometer. The mechanical loading platform can apply load ranging from 0.1 $mN$ to 500 $mN$, with accuracy better than 0.1 $mN$. The uniaxial load was applied using a flathead attached to the load cell. Load calibration was performed before each experiment. For tests at high temperature, one electronic heater and one RTD sensor were attached to both ends of the sample, for controlling and monitoring the temperature, respectively. Atomic force microscope cantilevers CT170 (Nanoscience Instruments, Inc., AZ) were used as the sample. It has the length of 225 $μm$, width of 40 $μm$, and thickness of 6.5 $μm$. Compared with other types of sample, e.g., whiskers and pillars, the cantilever has fine flat surfaces, which brought much convenience for the Raman spectroscopy measurement. Besides, the cantilevers are commercially available, which saved the effort for manufacturing samples with consistent dimensions. The cantilever was made of highly doped single-crystalline silicon. Before the measurement was performed, the cantilever was compressed using a trail load to ensure good contact between the load application module and the sample, which was represented by a linearly increasing loading curve. A trial load of 1–2 $mN$ was found to be enough to ensure good contact.

The laser used in this Raman spectroscopy setup was 514.5 $nm$ Ar + laser (Modu-Laser, Inc., UT). The laser was directed to the sample using single mode fiber, and then focused using a 40× objective ($NA$ = 0.65). The back-scattered laser was collected by the same objective and sent to the spectrometer (Acton SP2500, Princeton Instruments, Inc., NJ).

Deformation of the Cantilever Under Constant Load.

The thermal drift rate is a characteristic parameter of the system, and was assumed to be constant during the whole process of measurement. The thermal drift effect was treated by removing the thermal drift displacement from the whole mechanical loading curve. The correction of the thermal drift effect was applied as shown in Fig. 2(d). The thermal drift data acquired in the experiments (Fig. 2(c)) correspond to the red-circled part in Fig. 2(d). The thermal drift rate in Fig. 2(c) is only 0.0947 $nm/s$. However, it significantly alters the mechanical loading curve, as shown in Fig. 2(d). After compensation of the thermal drift, the corrected load–unload plot was obtained, shown as the black curve in Fig. 2(d), which is also the plot in Fig. 2(a).

Creep is the plastic deformation of solids at a constant stress level (less than yield stress) and temperature (lower than the melting temperature). Creep behavior of materials is usually studied under the condition of constant mechanical load and uniform temperature distribution. The creep behavior is usually described by Eq. (1). The strain rate $ɛ·$, and the stress $σ$ and are calculated in this research using the relations Eqs. (2) and (3); Ref. [11] Display Formula

(1)$ɛ·=Aσnexp(-QRT)$
Display Formula
(2)$ɛ·=dh(t)dt1h$
Display Formula
(3)$σ=F/As$

where $h(t)$ is the dimension of the sample along which the load is applied; $F$ is the applied load; and $As$ is the cross section of the sample. The stress exponent $n$ is calculated by taking logarithm to both sides of Eq. (1), which gives below equation: Display Formula

(4)$lnɛ·=nlnσ+ln(A·exp(-QRT))=nlnσ+C(T)$

where $C(T)$ is a constant at a specific temperature. If the $ɛ·$$σ$ curve is plotted in the double-log coordinates, the stress exponent $n$ will be the slope of the curve at the constant creep stage. As shown in Eq. (2), the derivation of the strain rate requires differentiation of the creep curve $h(t)$. However, the creep data are scattered dots, as illustrated in Fig. 2(b). In order to perform the differentiation, corresponding curve fitting is necessary to obtain a smooth function of the creep curve. A commonly used fitting function for $h(t)$ developed by Li and Ngan is given as below [24] Display Formula

(5)$h(t)=hi+a(t+ti)b+kt$

where $hi$, ti, $a$, $b$, and $k$ are fitting constants. The above equation was used by Ma et al. to investigate the creep behavior of thin Ni film [25], by Cao et al. in investigating the creep behavior of thin Ta film [13], and by Gan et al. to study the creep behavior of a Si-C-O based thermal barrier coating [11,26]. A slightly modified form (Eq. (6)) was used in this research for less fitting errors and better convergence [11]. Display Formula

(6)$h(t)=hi+atb+kt$

Using the $h(t)$ curve, and described Eqs. (2)(6), the creep strain rate and stress exponent can be calculated.

Stress Measurement by Raman Spectroscopy.

The Raman spectroscopy measurements to calculate surface stress was performed at the maximum load, when the load was held constant. The measurement of stress using Raman spectroscopy is based on the principle of inelastic interactions between the incident laser and the vibration of crystal lattice [27]. The lattice vibrations are quantized to different modes or phonons. In the case of silicon, the Raman modes consist of three degenerate $k=0$ optical phonon modes. For unstressed single-crystalline silicon at room temperature, these three modes have the same Raman frequency of $ω0$ at 520 $Rcm-1$ [27], which is determined by the natural lattice vibrations of this material. The stress-induced Raman shift is described by the Raman secular equation [28,29], given as below Display Formula

(7)$|pε11+q(ε22+ε33)-λ2rε122rε132rε12pε22+q(ε33+ε11)-λ2rε232rε132rε23pε33+q(ε11+ε22)-λ|=0$

Here, $p$, $q$, and $r$ are the optical phonon deformation potentials, with $p=-1.43ω02$, $q=-1.89ω02$, $r=-0.59ω02$ [28]; $εij$ are the strain tensor components; and $λ$ are the eigenvalues which are related to the Raman shift frequencies. When stress is applied to silicon, the Raman frequencies are related to the eigenvalues $λ$ as shown below Display Formula

(8)$λm=ωm2-ω02 (cm-2)$

here $ωm(m=1,2,3)$ is the Raman frequency when stress is applied. The Raman shift change $Δωm(m=1,2,3)$ is given as below [27,30] Display Formula

(9)$Δωm=ωm-ω0≈λm2ω0$

Since the strain tensor $εij$ and the stress tensor $σij$ are symmetric, the Hook's law is expressed as below Display Formula

(10)${ε}=[S]{σ}$

where ${ε}={ε11,ε22,ε33,2ε12,…}T$; ${σ}={σ11,σ22,σ33,σ12,…}T$; $S$ is the elastic compliance matrix. For a material with cubic crystal structure, there are three independent constants in the compliance matrix [31], which are expressed as shown in below equation Display Formula

(11)$[S]=[S11S12S120000S11S1200000S11000000S44000000S44000000S44](Pa-1)$

In the case of silicon, $S11=7.68×10-12 Pa-1$, $S12=-2.14×10-12 Pa-1$, and $S44=12.7×10-12 Pa-1$ [32]. For the measurement of mechanical stress inside silicon, the Raman shift difference $Δωm(m=1,2,3)$ is measured, and then it is related to the stress components $σij(i,j=1,2,3)$ by Eqs. (7)(11). In this research, the deformation of the silicon cantilever under constant load was investigated at 25 °C, 50 °C, and 100 °C, respectively. The compressive load at the holding stage was chosen to be 13 $mN$, 26 $mN$, and 39 $mN$. The corresponding compressive stress along the longitudinal direction of the cantilever was 50 $MPa$, 100 $MPa$, and 150 MPa. The mechanical loading and unloading rate were chosen to make the loading and unloading stages last for 10 s, respectively. The load was held at its maximum for 500 s to investigate the deformation at constant load. For each set of measurement, at least 10 repeated tests were performed in order to eliminate the effect of measurement error on physical quantities.

Results

In this section, the deformation of the silicon cantilevers as a function of time under constant load is analyzed first. The strain rate is analyzed as a function of both temperature and applied load. The results are compared with published research on silicon. The error in the experiments is also discussed.

Deformation of the Silicon Cantilever as a Function of Time.

As discussed before, the creep curve consists of three stages. The time frame of the second stage is much longer than that of the other two stages. The third stage of the creep curve is usually not of interest in experiments. For the silicon cantilever investigated in this research, the initial stage of the creep is very short (<2 s) in the observation time frame of 500 s. Therefore, the second stage of creep dominates the creep curve, which makes the creep curve follow a linear pattern, as shown in Fig. 3. The term $kt$ in Eq. (6) dominates the terms on the right side of the equation. At room temperature, the overall deformation from creep is limited (<30 $nm$), as shown in Fig. 3(a). As the temperature increases to 50 °C, the overall deformation increases dramatically by a factor of 5–6, as shown in Fig. 3(b). From 50 °C to 100 °C, the deformation of the cantilever under compression further increases, but the ratio of this increase is much lower than that from room temperature to 50 °C. At all temperatures, the deformation of the cantilever increases as a function of the compressive load.

Thermal Drift Correction and Temperature Control.

For the nanoscale measurements of displacement, thermal drift of the system plays an important role. It is caused by thermal expansion of the system due to uneven temperature distribution of different parts. In this research, the setup was maintained at a constant temperature before and during the measurements to ensure thermal equilibrium of the system. The whole equipment was placed inside a closed chamber to eliminate the air flow and other possible disturbance to the temperature distribution of the system. As shown in Fig. 4, temperature determines the magnitude of the thermal drift rate. The thermal drift rate increases when the temperature increases. Besides, at higher temperature, the variation of the thermal drift rate also increases, which means a higher fluctuation of the temperature distribution of the system. This phenomenon is common for high temperature tests. The temperature fluctuation of the system can be suppressed by minimizing the air flow inside the test chamber and also longer waiting time for the system to reach thermal equilibrium. Figure 4(b) shows the dependence of the thermal drift rate on the maximum load. It reveals that the thermal drift rate has low dependence on the maximum load at this loading range. After compensation of the thermal drift in the process described by Fig. 2, the realistic creep rate of the silicon cantilever was obtained.

Deformation Rate of the Cantilever as a Function of Load and Temperature.

At room temperature, the strain rate of the cantilever under uniaxial compressive load is in the order of $2×10-7s-1$, with slight increase as a function of maximum load, as shown in Fig. 5(a). After the holding time of 500 s, the overall deformation during the mechanical holding period corresponds to compressive strain of 0.01%. Figure 5(a) also shows that the strain rate increases dramatically as a function of temperature. From 25 °C to 100 °C, the strain rate increases by a factor of 10. Besides the dependence of the strain rate on temperature, the strain rate is also affected by the applied stress. Almost at all temperatures, the strain rate increases as a function of applied stress. The increase ratio of the strain rate from 50 $MPa$ to 150 $MPa$ is less than 2. From Fig. 5(a), it can be deduced that the strain rate of the silicon cantilever in this research is mainly affected by the temperature increase in the temperature range of 25 °C–100 °C, and in the stress range of 50 $MPa$–150 $MPa$.

The creep property of silicon has been investigated before. The comparison of the parameters of the sample and the testing conditions are listed in Table 1. The strain rate of the silicon cantilever from this research was also compared with literature values in Fig. 5(b). Most of the previous work was performed on bulk silicon in the temperature range of 600 °C–1300 °C. For silicon, the transition from brittle to ductile behavior happens in the temperature range of 520 °C–600 °C [23]. Therefore, the creep deformation of bulk silicon is unlikely to happen below 600 °C. However, for microscale silicon samples, the creep deformation occurs at relatively lower homologous temperature due to the size effect and surface stress.

In 1969, Myshlyaev et al. measured the creep curves of bulk scale single-crystalline silicon using uniaxial compression method [18]. They found that the steady state creep of silicon can be expressed by the kinetic equation Display Formula

(12)$ɛ·=1011s-1exp(-5.6 eV-2.7×10-21 cm3σkT)$

where $σ$ is the applied stress; $k$ is the Boltzmann constant; and $T$ is the absolute temperature. The equation is valid for stress level up to 100 $MPa$. According to this equation, the strain rate of bulk silicon has strong dependence on temperature and applied stress, as also shown in Fig. 5(b). Myshlyaev et al. also concluded that the steady state creep is controlled by barriers associated with dislocations with sub-boundaries [18]. Taylor et al. measured the creep of bulk silicon samples in the temperature range of 850 °C–1300 °C in 1972 and found that the logarithmic creep of the samples was observed when the temperature was below 1000 °C, and steady state creep was observed when the temperature was above 1000 °C [19]. The strain rate of the silicon sample they measured was in the range of the $3×10-7s-1$$3×10-6s-1$, which is comparable to the strain rate observed in this research, Fig. 5(b). The applied stress in Taylor's work was 1 $MPa$–7 $MPa$. Through the observation of transmission election microscope and chemical etch-pitting, they concluded that dislocation glide was the mechanism for both creep conditions, but the activation energy was lower at higher temperature. Walters et al. investigated the creep of bulk silicon through 4-point bending apparatus, and fitted the relationship between the strain rate and stress using Eq. (1); Ref. [20]. The activation energy they found is 224 $kJ/mol$, compared with 236 $kJ/mol$ obtained by Alexander et al. by experimentation above 1100 $K$ [17].

In Fig. 6, the strain rate is plotted as a function of temperature. In this research, the strain rate increases by a factor of 10 with 75 deg of temperature increase (25 °C–100 °C). This is for the condition of low homologous temperature (0.18–0.22) and microscale silicon sample. In contrast, at higher homologous temperature (>0.5), the transition from brittle to ductile behavior occurs. The dependence of creep rate on temperature increase is therefore much higher, Fig. 6(b). However, it should be remembered that the available data from other literature are for high homologous temperature and bulk silicon samples.

Stress Exponent.

Stress exponent $n$ in Eq. (1) has been used as an indicator of the creep mechanism of materials, especially metals. Similar analysis has been performed here and shown in Fig. 7(a). The stress exponent roughly exhibits an increase as a function of temperature. The slight decrease of the stress exponent from 25 °C to 50 °C is due to the measurement point of 150 $MPa$ at 50 °C. Overall, the stress exponent obtained in this research is relatively low (<1), which indicates that the creep behavior of this silicon cantilever at low homologous temperature cannot be explained by existing creep models.

The creep mechanism of silicon has also been analyzed earlier according to the Ashby map [33]. The Ashby map is based on experimental investigation of creep properties of silicon with the grain size of 100 $μm$. The main results in the map was conducted in the 1960s by Myshlyaev et al. [18]. If analyzed using the Ashby map, the homologous temperature range and the applied stress investigated in this research are categorized into the power-law creep. However, in the Ashby map, this region is not directly supported by experimental results.

In order to further investigate the fundamental relationship between the stress and strain rate, the surface stress of the silicon cantilever was measured using Raman spectroscopy with the spatial resolution of around 4 $μm$. The detail of the stress measurement is discussed elsewhere [34].

Error Analysis.

The creep measurement of the silicon cantilever requires accurate measurement of the displacement at micrometer or even nanometer scale. While compressive load is applied to the cantilever, it also compresses the sample holder. The overall displacement measured by the system may include the creep of the cantilever and the creep of the sample holder. For the cantilever used in this research, the cross section is $2.6×10-10 m2$, while the cross section of the cantilever base is larger than $7×10-7 m2$. The cross section of the sample holder is much bigger, which is in the order of $5×10-5 m2$. With the compressive load applied to the cantilever, the cantilever base and the sample holder, the deformation of the cantilever should dominate the overall deformation of the whole assembly, despite the material property of each part.

Another factor that could influence the accuracy of the creep measurement is the thermal expansion of different parts, especially for high temperature measurements. Although the creep of the sample holder does not play an important role, the thermal expansion and the thermal drift of the sample holder and other parts of the equipment could be critical, if not treated properly. In this research, the experimental setup was kept inside an enclosed chamber, where the inside temperature was kept constant with a temperature difference of about 3 °C above the outside temperature. The silicon cantilever and the end of the load application module were heated locally with electronic heaters. Thermal equilibrium was ensured by a constant reading from the RTD sensors attached to the sample and the load application module. This localized heating avoids disturbance to the functioning electrical parts of the test platform. However, it also creates a constant temperature gradient between the sample and other parts of the equipment. This temperature gradient may lead to extra thermal drift to the parts close to the sample. However, compared to global heating of the whole assembly, the localized heating method should introduce much less error to the measurement, because the global heating method will not only increase the thermal drift but also affect the functioning of the electric sensors of the system. An ideal case for high temperature measurement is localized heating to the sample, and perfect thermal isolation between the sample and any other part of the equipment. But in real practice, this is almost impossible. The best case would be localized heating of the sample and possible ways to minimize the heat flow from the sample to other parts of the system, e.g., heat shield and thermal barrier materials used in this research.

Discussion

Effect of Creep Deformation on Raman Spectroscopy Measurement.

Raman spectroscopy has been an effective tool to investigate the mechanical stress and strain of silicon at microscale [27,28,30]. It has been widely used for pointwise [30,35-37], line scanning [38,39], area mapping [40-45] of stress in silicon beams [30], thin films [35,39,40,42,44], cantilevers [43], and microstructures [38,41]. The measurement of stress using Raman spectroscopy is based on the principle of inelastic interactions between the incident laser and the vibration of crystal lattice [27].

The Raman secular equation relates the Raman shift to the strain of the silicon sample. The elastic matrix correlates the strain to the mechanical stress. Both relationships are one-to-one mapping. Actually, when the stress is constant, the strain is changing as a function of time, due to the creep effect; and when the strain is constant, the stress is changing as a function of time as well, due the relaxation effect. This phenomenon conflicts with the idea of stress or strain measurement using Raman spectroscopy. That means either the stress or the strain cannot be measured accurately from the Raman spectroscopy method. The Raman shift from a single time interval of 200 s is shown in Fig. 8(a). In this figure, the Raman shift values were linearly fitted as a function of time. The slope of the fitted line is $1.28×10-6 nm/s$, which means the Raman shift is barely affected by the creep effect during the measurement time. Considering the strain rate of the silicon cantilever is only $2.5×10-7 s-1$, for the time interval of 200 $s$, the strain change is $5×10-5$ or 0.005%, which is at least one order of magnitude lower than the strain level in the Raman spectroscopy measurement of this research. Therefore, in the temperature range of 25 °C–100 °C, and the stress level of tens to hundreds of $MPa$, the effect of creep on the Raman spectroscopy measurement is very limited during the exposure time of 200 s.

Correlation of Surface Stress and Creep Deformation.

For microscale or nanoscale structures, the atoms near the surfaces experience less bonding forces from neighboring atoms, compared with the inside atoms. This fact also applies to bulk materials. But at the bulk scale the effect is insignificant because of its atomic or nanometer range of scale. The lack of bonding force results in a “relaxed” state of the surface or near-surface atoms. Theoretically, the stress at free surfaces should be zero. Therefore, the stress at surface is zero and near the surface is lower than the bulk stress value. Besides, the stress also exhibits a depth dependency as the bonding force increases when the depth with regard to the surface increases.

Figure 8(b) shows the correlation between the surface stress and applied stress for the cantilever in this research. At room temperature, the surface stress roughly matches with the applied stress. At some points, the surface stress is slightly higher than the applied stress. When the temperature increases, the surface stress becomes lower than the applied stress, and it keep decreasing as temperature changes from 25 °C to 100 °C. This implies that the relaxation of the surface atoms increases as a function of temperature. In the meanwhile, the creep rate of materials also increases as a function of temperature increase. The mechanism of creep includes vacancy diffusion [2,3], grain boundary sliding [6], diffusion-controlled dislocation [7,8] and dislocation climb [9]. At micro/nanoscale, the material properties near the surface show significant size dependence. The relaxation of surface atoms will enhance the creep strain rate of the material. Across a given microscale cross section, surface atoms are likely to experience higher strain than the bulk atoms due to surface softening. This also explains that silicon can sustain more mechanical strain at microscale than bulk scale [46,47]. However, this does not mean the temperature dependency of the creep effect is entirely caused by the near-surface relaxation. The findings in this research only imply that the near-surface relaxation is contributing to the temperature dependency of the creep effect.

Conclusion

In this research, the creep behavior of micron sized silicon cantilever was investigated at the microscale and in the temperature range of 25 °C–100 °C. The stress level and the temperature range were chosen to correspond close to those in the semiconductor devices. The findings of this research are summarized as follows:

• At all the measurement temperatures, the creep of the silicon cantilever increases as a function of the mechanical stress in the range of 50 $MPa$–150 $MPa$.

• The creep rate of the silicon cantilever increases a function of temperature. But the increasing rate slows down with rise in temperature from 50 °C to 100 °C.

The findings in this research also reveal that the effect of creep on Raman spectroscopy measurement is insignificant, given the exposure time of the Raman spectroscopy measurement is not too long. This provides the basis for stress measurement of microscale silicon structures using Raman spectroscopy. With different laser wavelength, the penetration depth to silicon also varies. Therefore, the Raman spectroscopy method has the capability of investigating the depth sensitive stress distribution of stress inside silicon. Measurements of surface stresses revealed that at lower temperature surface stress value is close to the bulk value. However, as temperature increases, the surface stress values show a significant deviation from the bulk values.

Acknowledgements

Authors acknowledge support provided by Purdue Birck Nanotechnology Center. This work was supported by the United States National Science Foundation (US-NSF) Grant No. CMMI113112 1131112 (Program Manager: Dr. Dennis Carter).

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Yao, S. K., Xu, D. H., Xiong, B., and Wang, Y. L., 2013, “The Plastic and Creep Behavior of Silicon Microstructure at High Temperature,” The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII), Transducers & Eurosensors XXVII, Barcelona, Spain, pp. 159–162.
Huff, M. A., Nikolich, A. D., and Schmidt, M. A., 1993, “Design of Sealed Cavity Microstructures Formed by Silicon Wafer Bonding,” J. Microelectromech. Syst., 2(2), pp. 74–81.
Yasutake, K., Murakami, J., Umeno, M., and Kawabe, H., 1982, “Mechanical Properties of Heat-Treated CZ-Si Wafers From Brittle to Ductile Temperature Range,” Jpn. J. Appl. Phys., Part 1, 21(5), pp. 288–290.
Li, H., and Ngan, A. H. W., 2005, “Indentation Size Effects on the Strain Rate Sensitivity of Nanocrystalline Ni-25 at. %Al Thin Films,” Scr. Mater., 52(9), pp. 827–831.
Ma, Z. S., Long, S. G., Zhou, Y. C., and Pan, Y., 2008, “Indentation Scale Dependence of Tip-in Creep Behavior in Ni Thin Films,” Scr. Mater., 59(2), pp. 195–198.
Gan, M., and Tomar, V., 2011, “Scale and Temperature Dependent Creep Modeling and Experiments in Materials,” JOM, 63(9), pp. 27–34.
Wolf, I. D., 1996, “Micro-Raman Spectroscopy to Study Local Mechanical Stress in Silicon Integrated Circuits,” Semicond. Sci. Technol., 11(2), pp. 139–154.
Anastassakis, E., Pinczuk, A., Burstein, E., Pollak, F. H., and Cardona, M., 1970, “Effect of Static Uniaxial Stress on the Raman Spectrum of Silicon,” Solid State Commun., 8(2), pp. 133–138.
Ganesan, S., Maradudin, A. A., and Oitmaa, J., 1970, “A Lattice Theory of Morphic Effects in Crystals of the Diamond Structure,” Ann. Phys., 56(2), pp. 556–594.
Animoto, S. T., Chang, D. J., and Birkitt, A. D., 1998, “Stress Measurement in MEMS Using Raman Spectroscopy,” Proc. SPIE, 3512, pp. 123–129.
Nye, J. F., 1985, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University, Clarendon Press, Oxford, UK.
Wortman, J. J., and Evans, R. A., 1965, “Young's Modulus, Shear Modulus, and Poisson's Ratio in Silicon and Germanium,” J. Appl. Phys., 36(1), pp. 153–156.
Frost, H. J., and Ashby, M. F., 1982, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon, Oxford, UK.
Gan, M., and Tomar, V., 2014, “An in Situ Platform for the Investigation of Raman Shift in Micro-Scale Silicon Structures as a Function of Mechanical Stress and Temperature Increase,” Rev. Sci. Instrum., 85(1), p. 013902. [PubMed]
Kang, Y., Qiu, Y., Lei, Z., and Hu, M., 2005, “An Application of Raman Spectroscopy on the Measurement of Residual Stress in Porous Silicon,” Opt. Lasers Eng., 43(8), pp. 847–855.
Nolan, M., Perova, T., Moore, R. A., Moore, C. J., Berwick, K., and Gamble, H. S., 2000, “Micro-Raman Study of Stress Distribution Generated in Silicon During Proximity Rapid Thermal Diffusion,” Mater. Sci. Eng., B, 73(1–3), pp. 168–172.
Papadimitriou, D., Bitsakis, J., Lopez-Villegas, J. M., Samitier, J., and Morante, J. R., 1999, “Depth Dependence of Stress and Porosity in Porous Silicon: A Micro-Raman Study,” Thin Solid Films, 349(1–2), pp. 293–297.
Schmidt, U., Ibach, W., Muller, J., Weishaupt, K., and Hollricher, O., 2006, “Raman Spectral Imaging—A Nondestructive, High Resolution Analysis Technique for Local Stress Measurements in Silicon,” Vib. Spectrosc., 42(1), pp. 93–97.
Li, Q., Qiu, W., Tan, H., Guo, J., and Kang, Y., 2010, “Micro-Raman spectroscopy Stress Measurement Method for Porous Silicon Film,” Opt. Lasers Eng., 48(11), pp. 1119–1125.
Vetushka, A., Ledinský, M., Stuchlík, J., Mates, T., Fejfar, A., and Kočka, J., 2008, “Mapping of Mechanical Stress in Silicon Thin Films on Silicon Cantilevers by Raman Microspectroscopy,” J. Non-Cryst. Solids, 354(19–25), pp. 2235–2237.
Naumenko, D., Snitka, V., Duch, M., Torras, N., and Esteve, J., 2012, “Stress Mapping on the Porous Silicon Microcapsules by Raman Microscopy,” Microelectron. Eng., 98, pp. 488–491.
Amer, M. S., Durgam, L., and El-Ashry, M. M., 2006, “Raman Mapping of Local Phases and Local Stress Fields in Silicon-Silicon Carbide Composites,” Mater. Chem. Phys., 98(2–3), pp. 410–414.
Bauer, M., Gigler, A. M., Richter, C., and Stark, R. W., 2008, “Visualizing Stress in Silicon Micro Cantilevers Using Scanning Confocal Raman Spectroscopy,” Microelectron. Eng., 85(5–6), pp. 1443–1446.
Kouteva-Arguirova, S., Seifert, W., Kittler, M., and Reif, J., 2003, “Raman Measurement of Stress Distribution in Multicrystalline Silicon Materials,” Mater. Sci. Eng. B-Solid State Mater. Adv. Technol., 102(1–3), pp. 37–42.
Goodman, G. G., Pajcini, V., Smith, S. P., and Merrill, P. B., 2005, “Characterization of Strained Si Structures Using SIMS and Visible Raman,” Mater. Sci. Semicond. Process., 8(1–3), pp. 255–260.
Langdo, T. A., Currie, M. T., Lochtefeld, A., Hammond, R., Carlin, J. A., Erdtmann, M., Braithwaite, G., Yang, V. K., Vineis, C. J., Badawi, H., and Bulsara, M. T., 2003, “SiGe-Free Strained Si on Insulator by Wafer Bonding and Layer Transfer,” Appl. Phys. Lett., 82(24), pp. 4256–4258.
Urena, F., Olsen, S. H., Siller, L., Bhaskar, U., Pardoen, T., and Raskin, J.-P., 2012, “Strain in Silicon Nanowire Beams,” J. Appl. Phys., 112(11), p. 114506.
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References

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Cao, Z. H., Li, P. Y., Lu, H. M., Huang, Y. L., Zhou, Y. C., and Meng, X. K., 2009, “Indentation Size Effects on the Creep Behavior of Nanocrystalline Tetragonal Ta Films,” Scr. Mater., 60(6), pp. 415–418.
Mayo, M. J., and Nix, W. D., 1988, “A Micro-Indentation Study of Superplasticity in Pb, Sn, and Sn-38 wt. % Pb,” Acta Metall., 36(8), pp. 2183–2192.
Lucas, B., and Oliver, W., 1999, “Indentation Power-Law Creep of High-Purity Indium,” Metall. Mater. Trans. A, 30(3), pp. 601–610.
Asif, S. A. S., and Pethica, J. B., 1997, “Nanoindentation Creep of Single-Crystal Tungsten and Gallium Arsenide,” Philos. Mag. A, 76(6), pp. 1105–1118.
Alexander, H., and Haasen, P., 1986, “Dislocations in the Diamond Structure,” Solid State Physics: Advances in Research and Applications, F.Seitz, D.Turnbull, and H.Ehrenreich, eds., Academic Press, New York.
Myshlyaev, M. M., Nikitenko, V. I., and Nesterenko, V. I., 1969, “Dislocation Structure and Macroscopic Characteristics of Plastic Deformation at Creep of Silicon Crystals,” Phys. Status Solidi C, 36(1), pp. 89–96.
Taylor, T. A., and Barrett, C. R., 1972, “Creep and Recovery of Silicon Single Crystals,” Mater. Sci. Eng., 10, pp. 93–102.
Walters, D. S., and Spearing, S. M., 2000, “On the Flexural Creep of Single-Crystal Silicon,” Scr. Mater., 42(8), pp. 769–774.
Yao, S. K., Xu, D. H., Xiong, B., and Wang, Y. L., 2013, “The Plastic and Creep Behavior of Silicon Microstructure at High Temperature,” The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII), Transducers & Eurosensors XXVII, Barcelona, Spain, pp. 159–162.
Huff, M. A., Nikolich, A. D., and Schmidt, M. A., 1993, “Design of Sealed Cavity Microstructures Formed by Silicon Wafer Bonding,” J. Microelectromech. Syst., 2(2), pp. 74–81.
Yasutake, K., Murakami, J., Umeno, M., and Kawabe, H., 1982, “Mechanical Properties of Heat-Treated CZ-Si Wafers From Brittle to Ductile Temperature Range,” Jpn. J. Appl. Phys., Part 1, 21(5), pp. 288–290.
Li, H., and Ngan, A. H. W., 2005, “Indentation Size Effects on the Strain Rate Sensitivity of Nanocrystalline Ni-25 at. %Al Thin Films,” Scr. Mater., 52(9), pp. 827–831.
Ma, Z. S., Long, S. G., Zhou, Y. C., and Pan, Y., 2008, “Indentation Scale Dependence of Tip-in Creep Behavior in Ni Thin Films,” Scr. Mater., 59(2), pp. 195–198.
Gan, M., and Tomar, V., 2011, “Scale and Temperature Dependent Creep Modeling and Experiments in Materials,” JOM, 63(9), pp. 27–34.
Wolf, I. D., 1996, “Micro-Raman Spectroscopy to Study Local Mechanical Stress in Silicon Integrated Circuits,” Semicond. Sci. Technol., 11(2), pp. 139–154.
Anastassakis, E., Pinczuk, A., Burstein, E., Pollak, F. H., and Cardona, M., 1970, “Effect of Static Uniaxial Stress on the Raman Spectrum of Silicon,” Solid State Commun., 8(2), pp. 133–138.
Ganesan, S., Maradudin, A. A., and Oitmaa, J., 1970, “A Lattice Theory of Morphic Effects in Crystals of the Diamond Structure,” Ann. Phys., 56(2), pp. 556–594.
Animoto, S. T., Chang, D. J., and Birkitt, A. D., 1998, “Stress Measurement in MEMS Using Raman Spectroscopy,” Proc. SPIE, 3512, pp. 123–129.
Nye, J. F., 1985, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University, Clarendon Press, Oxford, UK.
Wortman, J. J., and Evans, R. A., 1965, “Young's Modulus, Shear Modulus, and Poisson's Ratio in Silicon and Germanium,” J. Appl. Phys., 36(1), pp. 153–156.
Frost, H. J., and Ashby, M. F., 1982, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics, Pergamon, Oxford, UK.
Gan, M., and Tomar, V., 2014, “An in Situ Platform for the Investigation of Raman Shift in Micro-Scale Silicon Structures as a Function of Mechanical Stress and Temperature Increase,” Rev. Sci. Instrum., 85(1), p. 013902. [PubMed]
Kang, Y., Qiu, Y., Lei, Z., and Hu, M., 2005, “An Application of Raman Spectroscopy on the Measurement of Residual Stress in Porous Silicon,” Opt. Lasers Eng., 43(8), pp. 847–855.
Nolan, M., Perova, T., Moore, R. A., Moore, C. J., Berwick, K., and Gamble, H. S., 2000, “Micro-Raman Study of Stress Distribution Generated in Silicon During Proximity Rapid Thermal Diffusion,” Mater. Sci. Eng., B, 73(1–3), pp. 168–172.
Papadimitriou, D., Bitsakis, J., Lopez-Villegas, J. M., Samitier, J., and Morante, J. R., 1999, “Depth Dependence of Stress and Porosity in Porous Silicon: A Micro-Raman Study,” Thin Solid Films, 349(1–2), pp. 293–297.
Schmidt, U., Ibach, W., Muller, J., Weishaupt, K., and Hollricher, O., 2006, “Raman Spectral Imaging—A Nondestructive, High Resolution Analysis Technique for Local Stress Measurements in Silicon,” Vib. Spectrosc., 42(1), pp. 93–97.
Li, Q., Qiu, W., Tan, H., Guo, J., and Kang, Y., 2010, “Micro-Raman spectroscopy Stress Measurement Method for Porous Silicon Film,” Opt. Lasers Eng., 48(11), pp. 1119–1125.
Vetushka, A., Ledinský, M., Stuchlík, J., Mates, T., Fejfar, A., and Kočka, J., 2008, “Mapping of Mechanical Stress in Silicon Thin Films on Silicon Cantilevers by Raman Microspectroscopy,” J. Non-Cryst. Solids, 354(19–25), pp. 2235–2237.
Naumenko, D., Snitka, V., Duch, M., Torras, N., and Esteve, J., 2012, “Stress Mapping on the Porous Silicon Microcapsules by Raman Microscopy,” Microelectron. Eng., 98, pp. 488–491.
Amer, M. S., Durgam, L., and El-Ashry, M. M., 2006, “Raman Mapping of Local Phases and Local Stress Fields in Silicon-Silicon Carbide Composites,” Mater. Chem. Phys., 98(2–3), pp. 410–414.
Bauer, M., Gigler, A. M., Richter, C., and Stark, R. W., 2008, “Visualizing Stress in Silicon Micro Cantilevers Using Scanning Confocal Raman Spectroscopy,” Microelectron. Eng., 85(5–6), pp. 1443–1446.
Kouteva-Arguirova, S., Seifert, W., Kittler, M., and Reif, J., 2003, “Raman Measurement of Stress Distribution in Multicrystalline Silicon Materials,” Mater. Sci. Eng. B-Solid State Mater. Adv. Technol., 102(1–3), pp. 37–42.
Goodman, G. G., Pajcini, V., Smith, S. P., and Merrill, P. B., 2005, “Characterization of Strained Si Structures Using SIMS and Visible Raman,” Mater. Sci. Semicond. Process., 8(1–3), pp. 255–260.
Langdo, T. A., Currie, M. T., Lochtefeld, A., Hammond, R., Carlin, J. A., Erdtmann, M., Braithwaite, G., Yang, V. K., Vineis, C. J., Badawi, H., and Bulsara, M. T., 2003, “SiGe-Free Strained Si on Insulator by Wafer Bonding and Layer Transfer,” Appl. Phys. Lett., 82(24), pp. 4256–4258.
Urena, F., Olsen, S. H., Siller, L., Bhaskar, U., Pardoen, T., and Raskin, J.-P., 2012, “Strain in Silicon Nanowire Beams,” J. Appl. Phys., 112(11), p. 114506.

Figures

Fig. 1

(a) A diagram of the experimental setup and (b) a detailed diagram of the mechanical loading and heating to the cantilever

Fig. 2

The mechanical loading process, combined with deformation measurement and thermal drift evaluation. (a) The overall load–unload curve; (b) the deformation as a function of time during steady state; (c) the thermal drift evaluation; and (d) the mechanical loading curve before and after correction of thermal drift

Fig. 3

Representative creep curves at (a) 25 °C; (b) 50 °C; and (c) 100 °C

Fig. 4

(a) Thermal drift as a function of applied load at different temperature and (b) thermal drift as a function of temperature under different loads

Fig. 5

(a) Strain rate of the silicon cantilever as a function of applied stress at 25 °C, 50 °C, and 100 °C and (b) comparison with literature values [19-21]

Fig. 6

(a) Strain rate of the silicon cantilever as a function of temperature and (b) comparison of results from this research and literatures [19-21]

Fig. 7

Stress exponent at 25 °C, 50 °C, and 100 °C

Fig. 8

(a) The effect of creep on Raman spectroscopy measurement and (b) comparison of near-surface stress and applied stress to the silicon cantilever

Tables

Table 1 A comparison of the parameters of the sample and test conditions with literatures
aSCS: single-crystalline silicon.

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